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arXiv:2406.11287 · Physical Review A

Precision measurement for open systems by the non-Hermitian linear response

Peng Xu, Gang Chen

Published Feb 14, 2025• 2 citations
arXivHTMLPDF

Abstract

The lower bound of estimated precision for a coherent parameter unitarily encoded in closed systems has been obtained, and such a lower bound is inversely proportional to the fluctuation of the encoding operator. In this paper, we first derive some general results regarding the lower bound of estimated precision for a dissipative parameter, which is nonunitarily encoded in open systems, by combining the law of error propagation and the non-Hermitian linear response theory. This lower bound is related to the correlation of the encoding dissipative operator and the evolution time. We next demonstrate the utility of our general results by considering three different kinds of nonunitary encoding processes, including particle loss, relaxation, and dephasing. We finally compare the lower bound with the quantum Fisher information obtained by tomography and find they are consistent in the regime where the non-Hermitian linear response applies. This lower bound can guide us to find the optimal initial states and detecting operators to significantly simplify the measurement process.

DOI
10.1103/physreva.111.022220

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The lower bound of estimated precision for a coherent parameter unitarily encoded in closed systems has been obtained, and such a lower bound is inversely proportional to the fluctuation of the encoding operator. In this paper, we first derive some general results regarding the lower bound of estimated precision for a dissipative parameter, which is nonunitarily encoded in open systems, by combining the law of error propagation and the non-Hermitian linear response theory. This lower bound is related to the correlation of the encoding dissipative operator and the evolution time. We next demonstrate the utility of our general results by considering three different kinds of nonunitary encoding processes, including particle loss, relaxation, and dephasing. We finally compare the lower bound with the quantum Fisher information obtained by tomography and find they are consistent in the regime where the non-Hermitian linear response applies. This lower bound can guide us to find the optimal initial states and detecting operators to significantly simplify the measurement process.
We investigate precision scaling at the interface between unitary and non-unitary encoding under generalized noise including single-particle and collective dephasing and decay. Using linear response theory and the error propagation formula, we derive analytic precision expressions for both the unitary parameter ΩΩΩ and the dissipation strength γγγ. For unitary encoding, when the observable commutes with a Hermitian noise operator, the optimal encoding time is independent of NNN, yielding the Heisenberg limit ΔΩ∝1/NΔΩ\propto 1 / NΔΩ∝1/N; otherwise the precision degrades to the standard quantum limit or ceases to improve with NNN. For non-unitary encoding, when [A^,O^]=0[\hat{A}, \hat{O}] = 0[A^,O^]=0, the precision is insensitive to intrinsic dynamics and encoding time, scaling as $Δγ\propto \sqrt{γ/ \expval*{\hat{L}^\dagger \hat{L}}}$. Notably, for collective decay, the Dicke state reaches the Heisenberg limit Δγ∝1/NΔγ\propto 1 / NΔγ∝1/N, demonstrating that entanglement can enhance non-unitary estimation. Our results provide a unified framework and practical guidance for designing quantum metrology protocols in noisy environments.
The precision of quantum parameter estimation is traditionally constrained by the quantum Cramér-Rao bound, which is based on the Hermitian measurement framework. Recent studies of non-Hermitian systems have suggested new possibilities for enhancing parameter-estimation sensitivity. Here, we experimentally realize quantum parameter estimation using a non-Hermitian observable on a linear optical platform. The parameter is encoded in single-photon probe states and read out with a Sagnac interferometer, which allows us to reconstruct the complex expectation value of the implemented non-Hermitian observable from interference fringes. We observe a reduced error-propagation variance compared with the optimal Hermitian observable for the same probe-state model. This advantage remains visible under amplitude-damping noise. We further analyze the complete optical measurement as a physical positive-operator-valued measure (POVM) and show, through the corresponding classical Fisher information (CFI), that the observed non-Hermitian advantage is consistent with the standard quantum metrological limit when all output ports are included. Our results provide an experimental route to non-Hermitian observable readout and clarify its operational meaning in quantum sensing.
The quantum Fisher information constrains the achievable precision in parameter estimation via the quantum Cramér-Rao bound, which has attracted much attention in Hermitian systems since the 60s of the last century. However, less attention has been paid to non-Hermitian systems. In this Letter, working with different logarithmic operators, we derive two previously unknown expressions for quantum Fisher information, and two Cramér-Rao bounds lower than the well-known one are found for non-Hermitian systems. These lower bounds are due to the merit of non-Hermitian observable and it can be understood as a result of extended regimes of optimization. Two experimentally feasible examples are presented to illustrate the theory, saturation of these bounds and estimation precisions beyond the Heisenberg limit are predicted and discussed. A setup to measure non-Hermitian observable is also proposed.
Quantum parameter estimation with Hermitian systems has been applied in various fields, but there are relatively few results concerning non-Hermitian systems. Here, we study the quantum parameter estimation for general non-Hermitian Hamiltonians and derive an intuitive expression of quantum Fisher information (QFI) for pure states. Furthermore, we propose the condition for optimal measurements, which is applicable to both Hermitian and non-Hermitian Hamiltonians. To illustrate these results, we calculate and study the QFI of a specific parity-time (PT)(\mathcal{PT})(PT) -symmetric non-Hermitian Hamiltonian and give the optimal measurement. Surprisingly, we find some interesting properties of this PT\mathcal{PT}PT-symmetric Hamiltonian QFI, such as the mutations in QFI at exceptional point. Moreover, we also compare the variance of estimation generated by the optimal measurement with the theoretical precision bound to verify the condition for optimal measurements we proposed.
Linear response theory and Green's functions provide a universal framework for understanding dynamical correlations in strongly correlated open quantum systems. While the theoretical foundation for non-Hermitian linear response has been recently established to describe dissipation and fluctuation-dissipation relations (FDR), generalizing these predictions onto practical quantum computers remains a formidable algorithmic challenge due to the intrinsically non-unitary nature of the dynamics. In this work, we present a systematic algorithmic framework that seamlessly transforms non-unitary multi-time correlation functions into a unitary form viable for digital quantum hardware. By mapping the vectorization of the Lindblad master equation into an expanded continuous-variable Liouville space, we employ the Schrödingerization technique to deterministically evaluate the non-Hermitian response. Furthermore, through hardware-aware simulations utilizing a 133-qubit device noise model, we demonstrate that our unitary framework robustly preserves the fundamental spectral information -- specifically the phase and oscillatory frequency -- against realistic depolarizing channels. By bypassing explicit non-unitary mid-circuit measurements, this approach intrinsically supports the integration of standard quantum error mitigation protocols, providing a scalable algorithmic blueprint for probing open-system universality on near-term hardware.
Linear response theory is the theoretical foundation of modern experiments. In particular, it plays a vital role in measuring quantum matters. Its main idea is to take the external probe signal of the physical system as a perturbation and use the correlation function in the unperturbed equilibrium state to depict the response to the observable in system. In recent half century, the linear response theory for the closed quantum system has achieved great success in experiments on quantum matters. In recent years, with the tremendous progress of the precise manipulation of the light-matter interaction, the ultracold atom experiments can precisely control dissipative quantum many-body systems. With the discovery of many novel phenomena, dissipative quantum systems and non-Hermitian physics have attracted extensive attention in theory and experiment. We developed a linear response theory, named non-Hermitian linear response theory, to deal with open quantum systems. This theory takes the non-Hermitian term and quantum noise, which are induced by dissipation, as an external perturbative input, to detect the properties of the quantum system, and relates the time evolution of the observable with the correlation function in the unperturbed state of the system. The non-Hermitian linear response theory provides a new method for distinguishing the exotic quantum phase from the normal phase. The theoretical predictions are highly consistent with the recent experimental results of cold atom systems. This paper will review the non-Hermitian linear response theory and discuss its applications in quantum many-body and time-reversal symmetric quantum systems.